Water is poured into a conical container (as shown in the diagram) of base radius $15$ cm and the

Question

Water is poured into a conical container (as shown in the diagram) of base radius $15$ cm and the semi-vertical angle of the cone is $\theta$, where $\sin \theta = 0.6$. The depth of water is $h$ cm. Apply the inflow-rate model in parts (iii) and (iv) after filling has begun, when the water depth is positive. Find the total height of the container. The volume of water is $V$ $\text{cm}^3$. Show that $V = 3\pi \left[ 500 - \frac{(20 - h)^3}{16} \right].$ Given that the water is poured at a rate of $kh^2$ $\text{cm}^3 / \text{s}$, where $k$ is a constant, find the rate of increase of $h$ in terms of $\pi$, $h$ and $k$. Given that the rate of increase of $h$ is $\frac{64}{9\pi}$ cm/s when $h = 10$ cm, find the value of $k$.

Accepted Answer

(i) $20\text{ cm}$ (ii) $V=3\pi\left[500-\dfrac{(20-h)^3}{16}\right]$ (iii) $\dfrac{\mathrm{d}h}{\mathrm{d}t}=\dfrac{16kh^2}{9\pi(20-h)^2}$ (iv) $k=4$

Differentiation - Connected Rates of Change

Differentiation - Connected Rates of Change

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